{\bf $τ$-Function Evaluation of Gap Probabilities in Orthogonal and Symplectic Matrix Ensembles}

Abstract

It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact $τ$-functions for certain Painlevé systems. We show that all exact evaluations of gap probabilities for classical orthogonal matrix ensembles, either known or derivable from the existing literature, are likewise $τ$-functions for certain Painlevé systems. In the case of symplectic matrix ensembles all exact evaluations, either known or derivable from the existing literature, are identified as the mean of two $τ$-functions, both of which correspond to Hamiltonians satisfying the same differential equation, differing only in the boundary condition. Furthermore the product of these two $τ$-functions gives the gap probability in the corresponding unitary symmetry case, while one of those $τ$-functions is the gap probability in the corresponding orthogonal symmetry case.